Friday, March 30, 2012

a net is a net is a net, i bet!

i'm currently LaTeχing a revision of a manuscript, and now there are problems with terminology ..

i just realised that i'm using the term "net" in two different senses:
  1. from topology, a net is a generalisation of a sequence: the index set no longer consists of integers, but an arbitrary directed set but not necessarily a poset.  (the usual example consists of subsets of a fixed set that are partially ordered by inclusion, but not necessarily totally so.)
  2. an ε-net (with ε positive), on the other hand, is a notion from metric geometry: roughly speaking, it is a locally finite approximation of a metric space.
it's not that much of an annoyance: for each ε-net of a doubling space, i'm constructing an approximation fε of a fixed Lipschitz function f.
in other words, i'm building a net from a net of nets!

sometimes we mathematicians have to be more inventive with terminology ..!

1 comment:

Leonid Kovalev said...

Not necessarily a poset?